Boundary integral equations are an efficient approach for the scattering of acoustic and electromagnetic waves from periodic arrays (gratings) of obstacles. The standard way to periodize is to replace the free-space Green's function kernel with its quasi-periodic cousin. This idea has two drawbacks: i) the quasi-periodic Green's function diverges (does not exist) for parameter families known as Wood's anomalies, even though the scattering problem remains well-posed, and ii) the lattice sum representation of the quasi-periodic Green's function converges in a disc, thus becomes unwieldy when obstacles have high aspect ratio.

We bypass both problems by means of a new integral representation that relies on the free-space Green's function alone, and adds auxiliary layer potentials on the boundary of the unit cell strip while expanding the linear system to enforce quasi-periodicity. The result is a (slightly larger) 2nd kind system that achieves spectral accuracy, is immune to Wood's anomalies, avoids lattice sums, handles large aspect ratios, and couples to existing scattering code. By careful summing of neighboring images, obstacles may intersect the unit cell walls.

If time, we will discuss a similar robust new approach to the photonic crystal band structure eigenvalue problem.

Joint work with Leslie Greengard (NYU).

This is a joint work with Chris Kurcz amd Lucas Monzon.

Bio:

George Biros holds Associate Professor appointments with the Schools of Computational Science and Engineering at Georgia Tech and The Wallace H. Coulter Department of Biomedical Engineering at Georgia Tech and Emory University. Prior to joining Georgia Tech, he was an assistant professor in Mechanical Engineering and Applied Mechanics, Bioengineering and Computer and Information Science at the University of Pennsylvania. He received his BS in Mechanical Engineering from Aristotle University Greece (1995), his MS in Biomedical Engineering from Carnegie Mellon (1996), and his PhD in Computational Science and Engineering also from Carnegie Mellon (2000). He was a postdoctoral associate at the Courant Institute from 2000 to 2003.

The fast multipole method (FMM) is a technique allowing the fast calculation of sums in O(N) or O(N ln N) steps with some prescribed tolerance. Original FMMs required analytical expansions of the kernel, for example using spherical harmonics or Taylor expansions. In recent years, the range of applicability and the ease of use of FMMs has been considerably extended by the introduction of black box or kernel independent techniques. In these approaches, the user only provides a subroutine to numerically calculate the kernel K. This allows changing the definition of K with practically no change to the computer program. In this talk we will present a novel method that attempts to address some of the shortcomings of existing kernel independent FMMs. This method is applicable to all kernels that are analytic in some appropriate region. An important property of the method is the fact that the multipole-to-local operator is diagonal, that is for p multipole coefficients, the cost of applying the operator is O(p). The computational cost is O(N) for non-oscillatory kernels and O(N ln N) for oscillatory kernels. The approach is based on Cauchy's integral formula. We will present a numerical analysis of the convergence, and methods to find the optimal parameters in the FMM. Numerical results will be presented for some benchmark calculations to demonstrate the accuracy as a function of the number of multipole coefficients, and the computational cost of the different steps.

We show that rotation can be carried out easily and stably through "pseudospectral" projection, without ever constructing the matrix entries themselves. In the simplest version of the method, projection is carried out on the equator of the rotated sphere. If only the lowest angular modes are required, the algorithm can be further accelerated by using a sequence of constant latitude circles.

This is joint work with Leslie Greengard.

Then I turn to the main topic of the talk – a method to enhance the efficiency of integral equation based schemes for elliptic PDEs on domains with corners, multi-wedge points, and mixed boundary conditions. The key ingredients are a block-diagonal inverse preconditioner 'R' and a fast recursion, 'i=1,...,n', where step 'i' inverts and compresses contributions to 'R' from the outermost quadrature panels on level 'i' of a locally 'n'-ply refined mesh. From an efficiency point of view, this method converts a non-smooth boundary into a smooth boundary and mixed boundary conditions into pure boundary conditions. The spectral properties of the system matrices in the linear systems that eventually have to be solved are essentially the same. The "corner difficulties" are gone.

The talk is available as a pdf-file:
http://www.maths.lth.se/na/staff/helsing/corners.pdf

The management of water resources is one of the big future challenges of our world. Besides the "big picture" how to provide the humankind with a sufficient amout of high quality potable water, there are a lot of problems to be solved on an operational level. For example, local water suppliers have to make sure that water is available when needed in sufficient quantity and quality. They have to deal both the requirement for the lowest utility prices and the fiscal requirements of their municipalities which might possibly be in a precarious financial situation. Usually, the energy expended for pumps in wells and pumping stations is their major cost factor. The complex rates of electricity providers have high-cost and low-cost periods as well as limitations of total energy consumption; violations result in considerable additional fees.

The goal of the project is to develop a software which is able to find an tradeoff between mimimum energy costs and the reduction of pump switches, where this last aspect minimizes service and maintenance efforts. The degree of freedom for the optimization is essentially an intelligent water storage management.

The project requires basically three steps for the team to be performed:

1. Modelling of water nets as well as the corresponding production and delivery problem, based on resources and demand forecasts
2. A concept how to deal conflicting goals, like cost optimization and reduction of pump switches
3. Definition and implementation of solution algorithms. This includes to decide which libraries of basic optimization algorithms should be used.

Besides mathamatics, it is intended to introduce some basic aspects of project management. For example, there will be assigned "roles" of the team members, like project manager, systems architect, or sales (I am the customer). A project schedule has to be set up in the beginning.

References:

G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, 1988

R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows - Theory, Algorithms, and Applications, Prentice Hall, 1993

http://zibopt.zib.de/ The optimization suite of the Zuse Zentrum Berlin is one possibility to attack the optimization problems of the project. The advantage is that it is free for academic purposes. However, the team members are free to choose any other tool for linear and integer optimization, provided they will have solved the licensing requirements. (I strongly recommend not to start from scratch, by implementing the Simplex algorithm.)

Prerequisites:

Methods of Linear Optimization as well as programming skills are a must (preferably in C/C++, especially if the team will decide to choose the ZIB Optimization Suite), experience in Discrete Optimization and Graph Theory a real advantage.

The cellular structure inside the human brain varies on the scale of micrometers, which is much smaller than the size of a voxel. There may be thousands of irregularly-shaped cells within a voxel, and they all contribute to the environment seen by water molecules whose displacement is measured by the MRI scanner. In the typical DMRI experiment, the time interval over which water diffusion is measured is in the range of 50-100 milliseconds. Using the diffusion coefficient of 'free' water at 37 degrees Celsius, D = 3e-9 m2/s, we get an estimated diffusion distance of 15-25 micrometers. Clearly, in a DMRI experiment, water molecules encounter numerous times inhomogeneities in the cellular environment, such as cell membranes, fibers, and macromolecules. We want to simulate the DMRI signal at the scale of a single voxel, while taking into account cellular structure and the shape and duration of the diffusion gradients.

I will describe the Bloch-Torrey partial differential equation which is a phenomenological equation that describes the time evolution of the nuclear magnetization in a sample. I will talk about the numerical solution of the Bloch-Torrey PDE using Green's functions and alternatively by finite elements discretization. Finally, the measured diffusion MRI signal is the sum of the magnetization in the sample and I give some analytical results on the aggregate signal under some simplying assumptions.

We will present our recent work in solving realistic multiscale system-level EM design simulation problems in both frequency domain and time domain. In frequency domain, we combine the spectral element method for coarse scales and finite element method for fine scales with a fast spectral integral method. In time domain, we hybridize the spectral element time-domain method and finite-element time-domain method together with the perfectly matched layer, with explicit and implicit time integration schemes for different subdomains. The discontinuous Galerkin method is used as the subdomain interface condition. In time domain, we further incorporate a nonlinear circuit solver, making it possible to perform nonlinear circuit simulation with RF interactions in a seamless manner. Several previously intractable multiscale problems will be illustrated.

Short-range interactions are straight forward to parallelize. We invoke multiple threads per compute core to alleviate partition and load imbalances. For the calculation of long-range interactions, we assign the multipole subtrees below a certain level to compute threads with affinity settings that conform to the interaction lists of the tree nodes and exploit the memory hierarchy.

On a Sun SunFire X4600 with 8 AMD Opteron 885 processors (16 cores) running at 2.6 GHz clock rate and 64 GB of memory, we observe a better than 15x speedup compared to the sequential version of FMM-Yukawa for two sample benchmark problems of 10 to 100 million charges uniformly distributed inside a unit box or on the surface of a unit sphere and require six-digit accuracy.

The contribution of the author is a new method for parallelizing the above fast algorithm on distributed-memory machines. The method assumes only a power-of-two number of processes and has been shown to strong scale up to thousands of cores of Blue Gene/P with 90% efficiency even for extremely small problems. This is achieved by carefully peeling off partitions of the frequency domain and applying them to the spatial domain as the butterfly algorithm progresses. The result is that, using 2^P processes in d dimensions, the only communication required by each process is P reduce-scatter summations over a team of 2^d processes. Results are presented for a generalized Radon transform in two-dimensions, though the implementation has been shown to work in arbitrary dimensions. The source code is available at http://code.google.com/p/butterflyfio.

Figure 1: Schematic cross-section of a PEMFC.

Figure 2: Experimental vs. theoretical water content (Image taken from Adam Z. Webers conference presentation).

Figure 3: Contact angle hysteresis (Image taken from Adam Z. Webers conference presentation).

Figure 4: Lattice Boltzmann simulation of liquid water flow through a porous medium (Movie obtained from Palabos image galery).

1. Fuel cell systems explained (2nd edition), Larminie, James; Dicks, Andrew, 2003 John Wiley & Sons
2. Wettability and capillary behavior of fibrous gas diffusion media for polymer electrolyte membrane fuel cells, J.T. Gostick et al., Journal of Power Sources 194/1(2009), dx.doi.org/10.1016/j.jpowsour.2009.04.052
3. Characterization of internal wetting in polymer electrolyte membrane gas diffusion layers, P. Cheung et al., Journal of Power Sources, 187/2, 2009, 487-492, dx.doi.org/10.1016/j.jpowsour.2008.11.036

1. The rotation of the Earth that generates a centrifugal force that reduces the gravity force,
2. The tides which change the distribution of the planet's mass and thus the values of gravitational forces.

1. Even the smallest external forces on aircraft mounted gravimeters,
2. The non-inertial nature of the reference frames that correspond to moving platforms which is due to the Earth's curvature.

1. Geodesy 3rd Edition, Torge, Wolfgang, 2001, de Gruyter, ISBN 3110170728
2. Gravitation, Misner, C., Thorne, K., Wheeler, J., 1973, Freeman, ISBN 0716703440
3. Satellite Geodesy 2nd Edition, Seeber, Gunter, 2003, de Gruyter, ISBN 3110175495

Keywords: Image stabilization, Image segmentation, Classification, Tracking, Particle filters.

Project Description:

The project consists in detecting moving objects in suburban and forest firing areas that could indicate people in danger. Automatic monitoring systems can help in this situation, either, making a fully automatic analysis or sending a pre-alarm, to indicate situations that require operator's attention. The data are video sequences acquired, supposedly, with an unmanned aerial vehicle (UAV), or remotely piloted vehicle.

There are several advantages of using UAV in fire monitoring with respect to manned planes (or helicopters); they have a relative the lower cost of operation; they can fly in dangerous situations; they can fly for longer period of time and they can be equipped with video cameras for performing autonomous scene analysis.

Challenges of the project are:

1. We have an unstable source video. The UVS motion introduces shifts and shakes of the scene in the video. Hence, a preprocessing for video stabilization could be useful [2].
2. The fire's smoke is moving and must not be confused with objects' motion. We will require classifying moving objects between smoke and interesting blobs. Color, texture and motion features can support such a classification [4,5].
3. Static objects in the scene, as trees or houses, can occlude the moving objects (MOs). Moreover smoke can partially or fully occlude such MOs. For managing this situation, a motion analysis could alert us for occlusions and allows tracking partially occluded object [3].
4. Noise may produce false positive. Thus, the motion analysis could provide us from the information for classifying among noise, smoke and MOs and then to eliminate false positives [5].

1. Handbook of Mathematical Models in Computer Vision, N. Paragios, Y Chen, O. Faugeras Eds. Springer NY, 2006
2. R. Szeliski, "Image Alignment and Stitching," Chap 17 in [1], pp 273-292.
3. A. Blake, "Visual tracking: a short research roadmap," Chap 18 in [1], pp 293-307.
4. M. Rivera, O. Ocegueda, J.L.Marroquin, "Entropy controlled quadratic Markov measure fields for efficient image segmentation," EEE Trans. on Image Process., vol 16.(12), 3047-3057, 2007.
5. C.M. Bishop Pattern Recognition and Machine Learning, Springer NY, 2006. In particular chapters: 7,9,12

Historically, randomized techniques have been less popular in numerical analysis. Most of them trade accuracy for speed, and in many numerical environments one does not want to add yet another source of inaccuracy to the calculation that is already sufficiently inaccurate. One could say that in numerical analysis probabilistic methods are an approach of last resort.

I will discuss several probabilistic algorithms of numerical linear algebra that are never less accurate than their deterministic counterparts, and in fact tend to produce better accuracy. In many situations, the new schemes have lower CPU time requirements than existing methods, both asymptotically and in terms of actual timings. I will illustrate the approach with several numerical examples, and discuss possible extensions.

Despite decades of zealous effort, there are remarkably few applications where fast integral equation solvers dominate. To help explain why, we will describe effective strategies, assess performance, and present remaining challenges associated with applying fast solvers to problems in interconnect model extraction, biomolecular electrostatics, bodies in microflows, nanophotonics, and Casimir force calculation.